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  1. Expectation Maximisation in Python: Coin Toss Example

    Expectation Maximisation with Python : Coin Toss

    This notebook implements the example, I consider a classic for understanding Expectation Maximisation.

    See: http://www.nature.com/nbt/journal/v26/n8/full/nbt1406.html

    Notations:

    \begin{align*} \theta_A &= \text{Probability of a Heads showing up given the coin tossed is A}\\ \theta_B &= \text{Probability of a Heads showing up given the coin tossed is B}\\ \end{align*}
  2. A frequent inequality

    $$ \frac{x-1}{x} \leq \ln(x) \leq x-1 \forall\ x>0 $$

    Consider \(f(x)=\ln(x)-\frac{x-1}{x}\)

    \(f'(x) = \frac{1}{x} - \frac{1}{x^2} = \frac{x-1}{x^2}\)

    Now consider the following two cases:

    Case A: \(0 < x \leq 1\) and Case B: \(1 < x < \infty ...

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  3. Distribution of one random variable less than other

    Problem

    \(P(Y<X)\) for any two independent random variables \(X,Y\)

    Solution

    We just follow the definition:

    \(P(Y<X) = \int_{-\infty}^{\infty}f_X(x)dx \int_{-\infty}^{x}f_Y(y)dy\)

    \(P(Y <X) = \int F_X(y)f_X(x)dx\)

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  4. Poisson Demysitified

    Learnt it the hard way. This one is more of a scratch pad, inspired from the previously discussed problme on estimating the size of restriction fragments.

    Notations:

    \(N_t\)= Number of events that have happened in the \(t^{th}\) time interval

    \(T\) = Time taken till the first arrival occurs(or the ...

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  5. Restriction Fragments, MLE, and Mixed Random Variables

    Let \(X_1, X_2, \dots , X_n\) be the lengths of \(n\) restriction fragments. Suppose that a biotechnique can measure fragment lengths accurately up to a given length c. That is, if \(X_i < c\), then the technique gives correct value \(X_i\) Show that MLE of \(\lambda = \frac{n-T_n}{S_n+cT_n}\) where \(S_n ...

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  6. Convolution Demysitifed

    Problem

    Given \(f(x) = \frac{x}{2}\) for \(0 \leq x \leq 2\). Find the pdf of \(x_1+x_2\) for \(x_1,x_2\) which are i.i.d.

    Wrong solution

    $$ f*g(t) = \int_{-\infty}^{\infty} f(w)g(t-w)dw $$

    Thus, blindly,

    \(f_S(s) = \int_{0}^s \frac{x(s-x ...

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  7. MATH 501 Project

    Introduction

    Model of a two state reparable system:

    $$ \begin{equation} \frac{dp_0}{dt} = -\lambda_0 p_0(t) + \int_0^1 \mu_1(x)p_1(x,t)dx + \int_0^1u^{*}(x,t)dx \end{equation} $$
    $$ \begin{equation} \frac{\partial p_1(x,t)}{\partial t} + \frac{\partial p_1(x,t)}{\partial x} = -\mu_1(x)p_1 ...
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  8. Runs in flips of a coin

    This problem happened to be in one of the screening examinations and is my favorite because it demonstrates an application of indicator variables

    Problem

    A run is defined as maximal subsequence of consecutive tosses all having the same outcome. So HHHTHHTTH has 5 runs.(HHH,T,HH,TT,H). Let ...

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  9. SVD v/s MDS v/s PCA

    Principle Component Analysis(PCA) is a relatively more famous than Singular Value Decomposition(SVD) or Multidimensional Scaling(MDS). When I was introduced to the latter two, I was utterly confused trying to figure out what goes in where.

    SVD

    Let \(X_{mxn}\) data matrix. For an easy to relate example ...

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  10. Multidimensional Scaling

    MDS is a statistical technique to visualize dissimilarity between points. The distances between two pointsin n-dimensions are visualized in 2 dimensions such that it represents the distance in n-dimensions as far as possible.

    It is important to note that, for MDS, we start of with a ‘distance’ matrix and not ...

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  11. Gumbel distribution expectation

    What

    If \(X_1,..,X_n\) is a random sample with density \(f(x;\theta)=e^{-(x-\theta)}e^{-e^{-(x-\theta)}}\) (\(x \in\mathbb{R}\)) and \(-\infty<\theta<\infty\), \(\quad\)i) Find the estimator of \(\theta\)

    Solution

    First let’s confirm if \(f(x;\theta)=e^{-(x-\theta)}e^{-e^{-(x- ...

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  12. BA = AB = I

    To Prove: If

    $$A_{nxn}$$

    and

    $$B_{nxn}$$

    such that AB=I, then BA=I

    $$AB=I \implies Rank(A), Rank(B)=n$$

    Reason: Rank(AB)

    $$\leq$$

    min(Rank A, Rank B)

    so B is a full rank matrix. Now consider B=BI

    $$\implies$$

    B-B(AB)=0

    $$\implies$$

    B-(BA ...

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  13. Proof for triangle inequality for case $x+y<0$

    \(-|x| \leq x \leq |x|\) and \(-|y| \leq y \leq |y|\) \(\implies\) \(-|x|-|y| \leq x+y \leq |x|+|y|\) \(\implies\) \(|x+y| \leq |x| +|y|\) for any real \(x,y\)

    The last implication comes from the fact: \(|x| \leq a \leftrightarrow -a \leq x \leq a\) for some \(a ...

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